U.U.D.M. Project Report 2008:15
Examensarbete i matematik, 30 hp
Handledare och examinator: Erik Ekström Augusti 2008
Stochastic Volatility and the Volatility Smile
Vassilis Galiotos
Department of Mathematics
Uppsala University
Uppsala University
Department of Mathematics
Stochastic Volatility and the Volatility Smile
Author: Vassilis Galiotos Supervisor: Erik Ekström
Abstract
The purpose of this project is to explain to some
extent the importance of stochastic volatility
models and implied volatility. The model that is
studied is the Heston model (1993). Our findings
confirm the common belief that the implied
volatility smile slopes downwards at the money if
the correlation between the spot returns and the
volatility is positive. Similarly, if the correlation is
negative the implied volatility slopes upwards.
Contents
Introduction………4
Chapter 1: From the Heston Model to the Heston PDE……….5
a. The Model……….5
b. Derivation of the PDE………...6
Chapter 2: Solving the PDE………...9
a. Partial Differential Equations and Numerical Methods……… 10
b. Some Finite Difference Approximations of the Derivatives……….10
c. Boundary Conditions……….12
d. Solution and Results………..13
Chapter 3: Implied Volatility and the Volatility Smile………17
a. The Importance of Implied Volatility………...17
b. Figures and Comments……….18
Chapter 4: Summary and Conclusions………23
References………25
Appendix………..27
Introduction
In 1973 F. Black and M. Scholes published a fundamental paper on option pricing, where they introduced the Black-Scholes equation and the Black-Scholes model [1].
The same year R. Merton [7] published a paper on the same topic independently. The paper of Black and Scholes had major effect on the world of finance since it gave an answer to the problem of pricing options.
Some of the important assumptions of the Black-Scholes model are that the underlying asset’s price process is continuous and that the volatility is constant. This last assumption would lead to the conclusion that if we plot volatility against the strike price we would obtain a straight line, parallel to the horizontal axis. Equalizing the Black-Scholes model with the market observed option price and solving for volatility gives us the implied volatility. However, when plotting implied volatility using real market data one typically obtains a convex curve, known as the “smile curve” or the “volatility smile”, with minimum price “at the money” i.e. where the strike price is equal to the underlying spot.
In order to have a more realistic approach to the problem of option pricing, jump models and stochastic volatility models have been introduced. Jump models deal with the assumption of continuity by allowing the spot asset’s process to jump1. When studying stochastic volatility models the volatility is described by a stochastic process. These models are used in order to price options where volatility varies over time. If we denote the underlying stock price by S, and , two wiener processes with correlation !, then S satisfies the stochastic differential equation
dS = rSdt + "(t)
where the volatility process "(t) satisfies some stochastic differential equation of the form
d(") = b(")dt + a(")
1. For more information on Jump Models see [3].
One of the most interesting papers on Stochastic Volatility models is that of Steven L. Heston [8]. Heston’s model allows the spot and the volatility processes to have positive, negative or zero correlation. In this project we investigate numerically what happens to the implied volatility curve in each of these three cases. Our results confirm that for zero correlation implied volatility is decreasing in the money and increasing out of the money. It obtains its minimum value at the money. The minimum point moves to the left for negative and to the right for positive correlation.
In the first chapter the Heston model and the Heston Partial Differential Equation (PDE) are presented. In the second chapter we will solve the PDE using numerical methods. In the third chapter we calculate implied volatilities for different stock prices and plot them using different correlations. Finally, chapter 4 summarizes the results.
Chapter 1: From the Heston model to the Heston PDE a. The model
Assume that the spot asset S at time t follows the diffusion
dS = µSdt + ,
where is a Wiener process. The volatility #(t) follows an Ornstein-Uhlenbeck2 process
,
where is another Wiener process, such that and are correlated with correlation !. Let and apply Ito’s formula in . The result is
d#(t) = [$% & 2'#(t)]dt + 2$ ,
2 See [4] Part V, 22, p239-240.
Then if we let ( = 2', , and " = 2$ we end up to the Heston model where
dS = µSdt + , (1)
d#(t) = ([) & #(t)]dt + " (2) (3)
The relationship between the parameter ) and the volatility #(t) determines the instantaneous drift of #. If )<# then the process decreases until the volatility goes under the theta parameter. Then it goes up again and so on. The parameter ) is the long-term variance. The parameter k shows how fast the process reverts to ). A high k implies higher rate of reversion and vice versa. The parameter " in (2) is the volatility of the volatility. Finally, the correlation between the two Wiener processes is denoted
by , where .
b. Derivation of the PDE
By looking at the model and comparing the number of the random sources (two Wiener processes) with the number of the risky traded assets (only the underlying spot since volatility is not traded) one can easily see that the Heston model is an incomplete model3. Therefore, it is not possible to obtain a unique price for any contingent claim using only the underlying asset and a bank account, which is normally the case for complete models such as the Black-Scholes model. In order for the portfolio to be hedged we need to have equal number of random sources with risky traded assets. What can be done in order to achieve this is to add a benchmark derivative to our portfolio which price we take as given and price uniquely every contingent claim in terms of that derivative. The market includes now equal risky traded assets and random processes. Then by denoting the portfolio as P and the relative weights of the bank account, the stock and the benchmark derivative as x, y, and z, respectively, we get
P = xB + yS + zC (4)
3 Meta-theorem 8.3.1 in [2] p. 118
where C = C(S,#,t) the benchmark derivative, for example a European call, and B is the value of a risk free asset, for example a bank account satisfying the differential equation
dB = rBdt (5)
An additional assumption is that P is self-financing and therefore4
dP = xdB + ydS + zdC (6)
Now we will try to hedge a contingent claim which in this case is denoted by U = U(S,#,t) using this self financing portfolio.
P = U (7) and therefore dP = dU (8),
and applying Ito’s formula in
U = U(S,#,t)
we get
(9)
where the index t, s and # denotes the derivative with respect to the indicated variable i.e.
4 See [2], Lemma 6.4, p.84.
Assuming that C is at least twice differentiable, using analogous notation and applying Ito’s formula on the benchmark derivative we get analogous result, i.e.
(10)
Substituting (1), (5) and (10) in (6) and separating the and terms we get
(11)
and since (8) holds the and dt terms in (9) and (11) must be equal. Thus by comparing the and terms the relative weights of the stock and the derivative are obtained. Particularly we find that
(12) and (13)
What is left is to compare the drifts and in order to do that we combine (4) and (7) and find that
xB = U – yS – zC (14)
and thus the drift in (11) becomes
which should be equal to
After substituting the weights and performing some manipulations to the aforementioned equality we end up with the relationship
(15)
where !(S,#,t) is the market price of volatility, and is independent of the derivative choice. After appropriate manipulations in (15) we obtain the partial differential equation
(16)
which is known as the Heston PDE. In the next chapter we will solve this PDE using numerical methods.
Chapter 2: Solving the PDE
In the previous chapter we derived the Heston PDE using the Heston model which is a stochastic volatility model. In order to solve this equation Steven L. Heston in [8]
proposed a “closed form solution”. Note that the closed form solution can be applied to the specific model but not to any stochastic volatility model. In this paper
numerical methods are used to solve the PDE. A short description of the method we use follows.
a. Partial Differential Equations and numerical methods
When we solve a PDE what we are trying to do is to find the value of the function, say u(x,t), that satisfies not only the given equation but also the boundary conditions.
Some of the most well-known and widely used methods are the finite difference method and the finite element method. The former is relatively simpler. We will use it to solve the Heston PDE. The solution is implemented in Matlab.
The concept behind the finite difference method is that one can discretize the problem domain in time and space creating a grid of mesh points in it. At this point the function u is an unknown function so we are not able to calculate the derivatives of u. What we do instead is to replace the first and second order derivatives by finite difference approximations (a brief presentation of some finite difference approximations follows). Using these approximations and the boundary conditions and stepping backward or forward in time we can approximate the value of u(x,t) at each of the grid points.
b. Some Finite Difference approximations of derivatives
Assume we choose a time step "t and space step "x in order to discretize our problem domain. We have and . The following approximations can be used to replace the first and second order derivatives.
Central difference:
Forward difference:
Backward difference:
For the second derivative we have
and finally for the first derivative with respect to time
The value of the option we want to price is known at time of maturity which we denote as T. Using this value and the other four boundary conditions and going step by step backward in time we can approximate the value of u at all the time levels until we reach t=0. In order to discretize the problem domain in time we can use one of the schemes that follow:
Implicit Euler scheme:
Explicit Euler scheme:
Crank-Nicolson scheme:
The Crank Nicolson scheme gives the most accurate solution however in order to keep the calculations simple we chose to work with the Implicit Euler scheme.
c. Boundary conditions
Before we start implementing our solution to Matlab, we ought to take a careful look at the boundary conditions. A boundary condition that specifies the value of the unknown function at the boundaries of the problem domain is known as Dirichlet boundary condition. Dirichlet boundary conditions are used at the points where we are at the time of maturity and where the spot price is equal to zero.
First we look at the time direction. Remember that the owner of a European call has the right to buy the stock at time of maturity at the price K. Thus the contract’s value at T is zero if the stock price is less than K and S-K otherwise (K-S for puts). The boundary condition at the point t=T is
.
When the stock is worthless it is safe to assume that the option is worthless.
The boundary condition we use at S=0 is
Neumann boundary conditions give the value of the derivative of the function at the boundaries. Such conditions are used at the points where the stock and the volatility take their largest values.
For large values of S the option price grows linearly. The boundary condition we use at the point S=Smax is
The option price is typically increasing with volatility. It is however bounded by the stock price. When the volatility obtains its highest value the option price tends to become constant. The boundary condition we use at this point is
The only boundary condition which seems quite complicated is the one at #=0, since we have to solve a PDE. Let’s take a closer look at it.
At #=0 we have
, (17)
This is exactly what happens to the Heston PDE if we set the volatility equal to zero.
Let us assume for simplicity that we have zero interest rate. Then the boundary condition becomes much simpler since the derivative with respect to the stock price vanishes and (17) becomes as follows:
After substituting the derivatives with forward difference approximations (see b.) and after appropriate manipulations we find that
This is what we are going to use for boundary condition at #=0. Alternative boundary conditions have been suggested for the volatility, as well as the spot direction (see [4]
p.242-243).
d. Solution and Results
After we have presented the finite differences with which we approximate the derivatives, the scheme we discretize in time and the boundary conditions we are ready to solve the equation. The Heston PDE has two space variables, the stock price S and the volatility #. It has first and second order derivatives with respect to these
two variables as well as a mixed derivative. We use central difference to approximate the first order derivatives. For the approximation of the mixed derivative we apply central difference with respect to both. Let us denote and . The mixed derivative becomes
If we substitute the derivatives with the approximations discussed above and after some rearrangements, the Heston PDE becomes
The index n refers to time, i to stock and j to the volatility. We can see that we use nine points of the n time level to calculate one point of the n-1 time level. We start at n=T we calculate the value of U at the previous time level, n=T-dt and stepping backwards step by step we will find the option price at present time.
In Figure 1 we see the plot of the option price for initial volatility #=0.4. In Figure 2 we can see the option price in three dimensions. The values of the parameters that are used are: *=30, "=0.25, !=0, )=0.4, +=0, Smax=90, Vmax=1, T=0.5, dt=0.001, k=2 and r=0. When we discretize the problem domain we use 30 steps in the spot direction and 80 steps in the volatility direction. Thus, the plot shows the arbitrage free price of a European call option expiring in 6 months, with strike price 30, and zero interest rate. For now we allow zero correlation between the underlying spot and the volatility process. Trying other correlation values, positive or negative had no observable effect in the option price curve.
Figure 1
Figure 2
In Figure 3 the option price is plotted against the volatility. The option price is increasing. Remember that at the highest volatility level the boundary condition is of the form . The effect of this condition is obvious at the #=#max boundary.
Figure 3
Chapter 3: Implied volatility and the Volatility Smile
a. The importance of implied volatility
As discussed in the introduction the Black-Scholes model ignores two possible behaviors of the stock process, and those are discontinuity of the stock process and changes in volatility. One way to handle the second one and to model our market in a more realistic way is to use stochastic volatility models such as the Heston model.
However the market still uses the Black-Scholes formula in order to price traded derivatives. The question is which value of volatility we should include in to the Black-Scholes formula in order to obtain the right option price.
After solving the Heston PDE we have calculated the option price in terms of an underlying asset. This asset has some volatility that varies over time. Therefore all we have to do to answer the question above is to equalize the price of the derivative we have calculated with the Black-Scholes formula and solve for the volatility. The solution to this equation will give us the implied volatility that corresponds to the option price that we calculated. In this sense we can say that the importance of the implied volatility is that it is that value of volatility which gives us a more accurate price of the derivative we are pricing.
In this section we will use the Financial Toolbox of Matlab in order to calculate implied volatilities. The Matlab function “blsimpv” returns the implied volatility values for the given parameters. The parameters that are required are S, K, r, time to maturity and the observed option price. Then we plot implied volatility against stock price and comment on the results.
We will investigate the role that the correlation between the two Wiener processes plays to the shape of the volatility curve. From [4] theorem 4.2, p. 288-289, for zero correlation we expect to find a smile curve, which is decreasing in the money and increasing out of the money. This means that a change in the strike price would have an effect in the implied volatility.
b. Figures and comments
In Figures 4 t0 10 we plot implied volatilities against the stock price for different correlations. In Figures 4-7 the parameters that are used are the same as the ones that were used while solving the PDE. For the next three changes in long term variance and initial volatility are made. Fixing time at t=0 so that we use the present option price and denoting the volatility as “,” the Black-Scholes formula becomes:
] (16)
where
and N is the cumulative distribution function of the standard normal distribution N(0,1).
As mentioned earlier (chapter 1, a) the parameter ) is the mean around which the volatility process fluctuates. We take initial volatility equal to ), i.e. #=0.4, and plot implied volatility. We observe that when the correlation between the volatility process and the spot returns is set equal to zero (see Figure 4), the curve is decreasing rapidly into the money, obtains its minimum price at the money and starts to increase again as it goes away out of the money. We see that we obtain the desirable smile shape.
Figure 4
In Figures 5 and 6 we allow positive correlation between the two Wiener processes.
We start by setting !=0.1, and we notice that the minimum has been slightly moved to the right and instead of being around S=30 i.e. at the money now it is closer to S=40. Increasing the correlation makes it easier to observe the change in the smile shape. In Figure 6 we set !=0.8. The curve now is continuously decreasing until approximately the point where the stock price is close to S=70. After this point it starts increasing.
Figure 5
Figure 6
Now we try setting negative correlation and we observe that the curve starts increasing before the point where we have equal values for stock and strike price.
Figure 7 shows that the minimum has now moved to the left and is close to the point where S=20.
Figure 7
In order to verify our results we will try the same plots with different parameters .We can for example reduce the squared long-term volatility in half, that is $=0.2, and run tests for zero, positive and negative correlation for initial volatility 0.2. As before the strike price is at K=30 and we start by setting zero correlation. Again we see that the implied volatility curve slopes downwards in the money and upwards out of the money.
Figure 8
Figure 9
For positive correlation, !=0.3, between the two wiener processes (Figure 9) the curve keeps decreasing until approximately the point where S=70.
Figure 10
Lastly in Figure 10, we set correlation equal to -0.1 and observe that the implied volatility curve starts increasing before the point where S=K.
Chapter 4: Summary and Conclusions
The paper of Black-Scholes in 1973 contributed enormously to the development of financial markets. The assumption however of constant volatility has been a drawback in the modeling of the markets. Studying stochastic volatility models can be very useful since they capture the, more likely to occur, situation where volatility varies over time. In this thesis the Heston model, one of the most popular stochastic volatility models, was studied. After solving the Heston PDE using the Finite Difference Method we obtained the arbitrage free price of a European call option. We used this price to calculate implied volatilities by setting the option price equal to the Black-Scholes model and solving for volatility. Our results showed that different
behaviors of the implied volatility curve are obtained depending on the sign of the correlation. For zero correlation the volatility decreases in the money and increases out of the money. Setting positive correlation changed the shape of the smile. We observed that the smile sloped downwards at the money. The opposite result occurred for negative correlation. In all cases a smile was obtained.
References
[1] BLACK F. and SCHOLES M. , (1973): “The Valuation of Options and Corporate Liabilities”, Journal of Political Economy, Vol.81, p.637-659.
[2] BJÖRK T. (2004): Arbitrage Theory in Continuous Time. Oxford University Press, Second Edition.
[3] CONT RAMA and TANKOV PETER (2003): Financial Modeling With Jump Processes. Chapman & Hall/CRC Financial Mathematics Series.
[4] DUFFY, DANIEL J. (2006): Finite Difference Methods in Financial Engineering.
A Partial Differential Equation Approach. John Wiley & Sons, Ltd.
[5] FOUQUE JEAN-PIERRE, PAPANICOLAOU GEORGE and SIRCAR RONNIE K. (2000): Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press.
[6] HEATH MICHAEL T. (2002): Scientific Computing. An Introductory Survey.
McGraw Hill, Second Edition.
[7] MERTON, R. (1973) “Theory of Rational Option Pricing”, The Bell Journal of Economics and Management Science, Vol.4, p.141-183.
[8] STEVEN L. HESTON (1993): “A Close-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”. The Review of Financial Studies, Vol. 6, p. 327-343.
[9] HULL, J. and WHITE, A. (1987): “The Pricing of Options on Assets with Stochastic Volatilities”. The Journal of Finance, Vol. 42, p.281-300
[10] KLUGE, T. (2002): “Pricing Derivatives in Stochastic Volatility Models using the Finite Difference Method”. Diploma thesis, Technical University, Chemnitz.
[11] RENAULT, E. and TOUZI, N. (1996): “Option Hedging and Implied Volatilities in a Stochastic Volatility Model”. Mathematical Finance, Vol. 6, p. 279-302.
[12] WANG, J. (2007): “Convexity of Option Prices in the Heston model” Project Report, Uppsala University, Uppsala, Sweden.
Appendix
1.Matlab code for the option price
sigma=0.25;
rho=0.0;
k=2;
r=0;
theta=0.4;
lamda=0;
smax=90;
vmax=1;
dt=0.001;
T=0.5;
K=30;
m=30;
n=80;
ds=smax/m;
dv=vmax/n;
u=zeros(m+1,n+1,T/dt+1);
for i=1:m+1 for j=1:n+1 for t=T/dt+1 if (i-1)*ds<K u(i,j,t)=0;
else
u(i,j,t)=(i-1)*ds-K;
end end end end
for t=T/dt:-1:1 for i=2:m
u(i,1,t)=(1-k*theta*dt/dv)*u(i,1,t+1)+(k*theta*dt/dv)*u(i,2,t+1);
end
for i=2:m for j=2:n
u(i,j,t)=dt*(u(i,j,t+1)*(1/dt-(j-1)*dv*((i-1)*ds)^2/(ds)^2-(j-
1)*dv*sigma^2/(dv)^2-r)+u(i+1,j,t+1)*((j-1)*dv*((i-1)*ds)^2/(2*ds^2)+r*(i- 1)*ds/(2*ds))+u(i-1,j,t+1)*((j-1)*dv*((i-1)*ds)^2/(2*ds^2)-r*(i-
1)*ds/(2*ds))+u(i,j+1,t+1)*(sigma^2*(j-1)*dv/(2*dv^2)+(k*(theta-(j-1)*dv)- lamda)/(2*dv))+u(i,j-1,t+1)*(sigma^2*(j-1)*dv/(2*dv^2)-(k*(theta-(j-1)*dv)- lamda)/(2*dv))+rho*sigma*(i-1)*ds*(j-1)*dv/(4*ds*dv)*(u(i+1,j+1,t+1)-u(i- 1,j+1,t+1)-u(i+1,j-1,t+1)+u(i-1,j-1,t+1)));
end end for j=1:n
u(m+1,j,t)=u(m,j,t)+ds;
end
for i=2:m+1
u(i,n+1,t)=u(i,n,t);
end end i=1:m+1;
j=1:n+1;
plot((i-1)*ds,u(:,33,1)); % the option price;
%plot((i-1)*ds,u(:,33,T/dt+1)); %the payoff;
%mesh(u(:,:,1)); % the option price 3D;
%plot((j-1)*dv,u(10,:,1)); % the option price against volatility;
2.Matlab code for the volatility smile.
sigma=0.25;
rho=0.0;
k=2;
r=0;
theta=0.4;
lamda=0;
smax=90;
vmax=1;
dt=0.001;
T=0.5;
K=30;
m=30;
n=80;
ds=smax/m;
dv=vmax/n;
u=zeros(m+1,n+1,T/dt+1);
for i=1:m+1 for j=1:n+1 for t=T/dt+1 if (i-1)*ds<K u(i,j,t)=0;
else
u(i,j,t)=(i-1)*ds-K;
end end end end
for t=T/dt:-1:1 for i=2:m
u(i,1,t)=(1-k*theta*dt/dv)*u(i,1,t+1)+(k*theta*dt/dv)*u(i,1,t+1);
end for i=2:m for j=2:n
u(i,j,t)=dt*(u(i,j,t+1)*(1/dt-(j-1)*dv*((i-1)*ds)^2/(ds)^2-(j-
1)*dv*sigma^2/(dv)^2-r)+u(i+1,j,t+1)*((j-1)*dv*((i-1)*ds)^2/(2*ds^2)+r*(i- 1)*ds/(2*ds))+u(i-1,j,t+1)*((j-1)*dv*((i-1)*ds)^2/(2*ds^2)-r*(i-
1)*ds/(2*ds))+u(i,j+1,t+1)*(sigma^2*(j-1)*dv/(2*dv^2)+(k*(theta-(j-1)*dv)- lamda)/(2*dv))+u(i,j-1,t+1)*(sigma^2*(j-1)*dv/(2*dv^2)-(k*(theta-(j-1)*dv)- lamda)/(2*dv))+rho*sigma*(i-1)*ds*j*dv/(4*ds*dv)*(u(i+1,j+1,t+1)-u(i-1,j+1,t+1)- u(i+1,j-1,t+1)+u(i-1,j-1,t+1)));
end end
for j=1:n
u(m+1,j,t)=u(m,j,t)+ds;
end
for i=2:m+1
u(i,n+1,t)=u(i,n,t);
end end
F=zeros(m+1,1);
for i=1:m+1
F(i,1)=blsimpv((i-1)*ds,K,r,T,u(i,33,1));
end i=1:m+1;
plot((i-1)*ds,F); % the volatility smile;
